Physical characterization of late-type contact binary systems observed by LAMOST: a comprehensive statistical analysis

This paper presents a catalog of approximately 1800 Eclipsing W UMa systems (EWs) using parameters from LAMOST, VSX, ZTF and Gaia. Our detailed statistical analysis includes frequency distributions of parameters, confidence intervals, and hypothesis testing to provide deeper insights into the physical properties of this important eclipsing binary class. We focus on key parameters, including Period, Effective Temperature, Surface Gravity, metallicity, Radial Velocity, and spectral type of the systems. Our study reveals that the mean values for period, effective temperature, logarithmic surface gravity, metallicity, and radial velocity for EW systems are 0.377 days, 5775 K, 4, -0.185, and -4.085 km/s, respectively. The 95% confidence intervals for these parameters are 0.372 to 0.382 days, 5730 to 5820 K, -0.202 to -0.168, 3.97 to 4.03, and -6.47 to -1.7 km/s, respectively. Hypothesis testing of the estimated intervals results in the acceptance of the null hypothesis, indicating that EW systems are characterized within the specified limits. Our study also confirms that the majority of EW systems are late-type stars, primarily classified as F spectral type, followed by G and K. Interestingly, among the sample, 88 systems are classified as A spectral type, with a mean surface temperature of 7400 K. We examine the correlation between orbital periods and atmospheric parameters in the VSX and ZTF catalogs. While ZTF periods align well with established relations (correlation coefficient: 0.74), a weaker correlation is found in the VSX catalog. This highlights the need for a revision of VSX periods for improved accuracy in the studied sample of EWs.


Data
We gathered our sample data from the LAMOST DR7 V2.0 [http:// dr7.lamost.org/] catalogue and conducted a cross-match with the VSX (Variable Star Index), ZTF (Zwicky Transient Facility) variable star catalog ( 30 ) and GAIA DR3 (Global Astrometric Interferometer for Astrophysics Data Release3) to determine the period, system IDs and distance (in Kpc) of the stars in our study.The criterion for identification involved ensuring that this offset was less than 2 arcseconds.For this investigation, we specifically selected the LAMOST LRS Stellar Parameter Catalog of A, F, G, and K Stars, which is expected to encompass the EW systems of interest.The LAMOST, also known as the Guoshoujing Telescope, is a remarkable 4-meter quasi-meridian reflecting Schmidt telescope equipped with 4000 fibers, allowing simultaneous spectroscopic observations within its expansive 5° field of view.Notably, starting in 2017, new medium-resolution spectrographs with a resolving power of R = 7500 were incorporated alongside the existing low-resolution spectrographs (R = 1800) 31 .Atmospheric parameters and spectral classes are determined for the observed objects automatically by LASP (LAMOST stellar parameter pipeline) 32 .This automated process relies on the Universite de Lyon spectroscopic analysis software (ULySS) developed by 33 .Utilizing empirical spectral libraries, such as ELODIE, and an implemented interpolator function called TGM ( 34,35 ), ULySS accurately fits the whole observed spectra.According to 32 , the intrinsic external accuracies derived for high-quality AFGK stellar spectra using ULySS are 43 K, 0.13 dex, and 0.05 dex for Teff, log g, and [Fe/H], respectively.The spectra are selected with the criterion of S/N in g band < 6 in dark nights, and S/N in g band < 15 in bright nights (see, 36 ).
]).The table provides essential details such as system names, types of light curve variability, spectral type, angular separation in arcseconds (between LAMOST and VSX), LAMOST observing date, right ascensions (RA), declinations (Dec), orbital periods per day, effective temperature, log of surface gravity, metallicity, radial velocity, as well as parallax and proper motion with their respective errors.For access to the complete version of the table, please refer to https:// zenodo.org/ record/ 84326 15.
Table 2 presents key statistical parameters for our sample.The mean period is approximately 0.377 with a standard error of 0.003, and the range for this parameter spans from 0.187 to 0.798.The effective temperature ( T eff ) has a mean value of about 5770 K with a standard error of 20 K, and its range extends from 3860 to 8360 K. Log(g) or Surface gravity's mean value is approximately 4, with a standard error of 0.017, and the range for this parameter ranges from 0.117 to 4.865.The dataset's mean metallicity ([Fe/H]) is around − 0.185 with a standard error of 0.009, and its range spans from − 2.273 to 0.566.Finally, the mean radial velocity (RV) is approximately − 4 km/s with a standard error of 1.221, and the range for this parameter is from − 395 to 284 km/s.

Method
Our method for constructing the statistical study of the physical parameters under investigation involves the following steps: 1. Range Calculation: First, we determine the range (R) of the dataset.This is achieved by finding the difference between the maximum and minimum values.2. Interval Determination: To establish the number of intervals (n) for the frequency distribution, We adopt Sturges's rule.This rule is expressed by the equation: where N represents the total number of data points in the dataset.
3-Interval Length Computation: With the number of intervals (n) determined, we proceed to calculate the interval length (L).This is accomplished using the formula: (1) where R denotes the range obtained in the first step.By following these steps, we effectively organize the dataset into a meaningful frequency distribution, shedding light on the distribution and variability of the physical parameters.Parameters listed in Table 2 such as the sample size (N), mean (x) , standard deviation (σ ) , minimum, maximum values, and the computed range (R) using our method, are used for this purpose.Details about the method can be found at 38 .

Frequency distribution
In this subsection, we applied Eqs. ( 1) and ( 2) to obtain the number of intervals (n) and the interval length (L) for each parameter in our sample, resulting in 12 intervals for the frequency distribution.The details of the distributions are presented in the following tables.

Period
Table 3 illustrates the distribution of the "Period" parameter in our sample.Notably, there are 3 instances with a period value of 0.187, while the majority of data points lie above 0.2.The EW period is highly concentrated within the range of 0.2 to 0.493 (periods less than 0.5), accounting for 1563 data points or approximately 87.8% of the dataset.Furthermore, periods less than 0.6 constitute 1664 data points, representing 93.6% of the dataset.However, in the last four intervals from 0.6 to 0.8, there are only 114 EW cycles, making up approximately 6.4% of the dataset.This finding indicates a significant concentration of the EW orbital period between 0.2 and 0.6, with occurrences above 0.6 being minimal.The corresponding graph (see Fig. 1a) visually illustrates this distribution pattern.
Effective temperature ( T eff ) In the T eff Frequency Distribution" Table (4), we observe significant insights regarding the concentration of temperature degrees ( T eff ) within the EW sample.Notably, a considerable proportion of T eff values, totaling 1619 instances, fall within the range of 4236 to 7236, representing approximately 91% of the dataset.On the other hand, the interval from 7236 to 8362 contain a smaller count of only 96 EW systems, accounting for approximately 5.4%.This finding highlights the predominant occurrence of EW temperature degrees between 4236 and 7236, indicating a well-defined concentration in this range.To visually illustrate this distribution pattern, we provide the accompanying figure depicting (Fig. 1b) the frequency distribution of EW temperatures (Table 4).

Surface gravity (Log(g))
In the "Log(g) Frequency Distribution" Table 5, it is evident that the highest concentration of log(g) values occurs between 3.68 and 4.87, with a total count of 1524, accounting for 85.6% of the dataset.The remaining 257 instances, comprising 14.4% of the data, are distributed across the other nine intervals.This finding underscores the dominant occurrence of log(g) values between 3.68 and 4.87, as illustrated in Fig. 1c).

Metallicity ([Fe/H])
The [Fe/H] frequency distribution, Table 6 provides significant insights into the distribution of metallicity values within the EW sample.Notably, the largest distribution of EW systems falls within the range from − 0.14 to 0.097, with a total of 612 systems representing 34.4% of the dataset.This observation indicates that the majority of EW systems in the sample are old stellar population, reflecting their prevalence in the specified metallicity range.
(2) It is important also to highlight that, RV were observed at different phases and are varying with time.0][41][42] ) show that the secondary component exhibits a higher radial velocity than the primary one.This means that in our sample, the higher RV values (e.g.283 and − 396) can be explained by observing systems' secondary component.Figure1e visually illustrates this distribution pattern, providing additional clarity on the prevailing trends of radial velocities in our sample.

Spectral types
As mentioned above, Our sample comprises different spectral type including A, F, G and K in this section we are aiming to understand their frequency distribution in addition to the physical properties of each type from the database.Statistical analysis of A spectral type.The binarity of early type stars including A-type by using Lamost is discussed by 43 and more recently by 44 .They reported that the binary fraction is decreases toward A-type stars.The detection of EWs with A-type stars is not common compared with the later spectral types (i.e.F, G and K) 45,46 .The present sample is originally contains about 21850 A-type stars, only 88 of them are found to be W UMa binaries.Referring to Table 8 for the A spectral type, a notable trend emerges: A significant 44.3% of the observed EWs fall under the category of A7V, amounting to 39 instances.Similarly, A6IV claims 20 occurrences, accounting for 22.7% of the total.These two categories collectively exert a substantial influence, commanding a combined ratio of 67% within this type.
Presented within this table are the comprehensive descriptive statistics encompassing all parameters specific to type A, comprising a total of 88 EWs instances.Notable observations include the mean period, hovering around 0.375.Comparatively, upon cross-referencing with Table 2, it is apparent that this value remains largely consistent across the spectrum, indicating a uniform mean period across all EWs spectral types.Interestingly, and as listed in Table 9, the mean temperature attains a notably higher value of 7400, which naturally corresponds to the youthful nature of these diminutive stars, characterized by temperatures spanning 7500 to 10,000 K in accordance with the Harvard classification.As for the mean log(g) (4.12), its proximity to the overall sample average of approximately 4 reinforces the coherent tendencies observed throughout the sample.Noteworthy, the mean metallicity registers at − 0.337, contributing an insightful marker of this type's elemental composition.In the realm of motion, the average radial velocity assumes a value of − 1.5, whereby the negative sign signifies a pronounced blue shift.
Statistical analysis of F spectral type.The detection of F-type is believed to be common toward EWs as reported by 47 .They reported that among 90 EWs, 52 systems are classified as F-type.Our results that listed in Table10, indicate that a substantial portion of the EWs population resides within the F0 spectral type, amounting to 239 instances and constituting 40.7% of the dataset.Likewise, F5 captures a significant share of 18.7%, encompassing a total of 128 EWs.In tandem, these two spectral types collectively contribute to an approximate total of 59.4%, highlighting their considerable prevalence among the observed EWs.
In the F-type stars, as depicted in Table 11, several noteworthy patterns emerge.The mean period closely approximates the overall mean found in the collective sample encompassing all spectral types.The average temperature, quantified at 6473.6, aligns remarkably well with the Harvard classification's reasonable range of 6000-7500 K for this type.The mean gravitational acceleration (log(g)) tends to converge towards the overall sample average of 4. Meanwhile, the mean metallicity registers at − 0.191, surpassing the overall sample mean.Notably, the mean radial velocity (RV) stands at − 7.32, underscoring a distinct propensity towards a blue shift for the majority of EWs within this category.However, the number of detailed spectroscopic study and RV curves of EWs remains small compared to the known EWs and even compared with our sample, the more recent sample introduced by 14 exhibiting an average RV of ∼ 7.7 km/s with red-shifted.This means that more observations are necessary to better understand the RV nature of EWs.
Statistical analysis of G spectral type.In the G spectral type Table 12, it becomes evident that the spectral categories G2, G5, G3, G7, and G8 collectively make up a significant portion, amounting to 74.2% of the dataset and totaling 498 instances of EWs.
Within the spectral type G Table 13 a noteworthy observation emerges: the mean values for period, log(g), and metallicity closely align with the overall sample average.Nonetheless, a distinction arises in terms of radial velocity, deviating from the norm and registering at -1.96.Concurrently, the mean temperature attributed to  www.nature.com/scientificreports/stars within this spectral type rests at 5408.This value aptly situates itself within the Harvard classification range of 5200-6000, affirming the consistent and accurate spectral classifications for stars within this category.
Statistical analysis of K spectral type.The spectral type K listed in Table 14 reveals a total count of 336 EWs across various spectral subtypes, excluding K6, K8, and K9.Remarkably, the majority of EWs instances are concentrated within K3, K5, and K7, amassing to a substantial 231 occurrences, constituting a significant 68.75% of the total count.Within the K-spectral type (see Table 15), we observe that the average values for period and log(g) closely align with the overall sample mean across various spectral types.However, there are notable distinctions in terms of metallicity and radial velocity, registering at -0.144 and -2.4,respectively.The mean temperature within this classification hovers around 4644, effectively situating it within the 3700-5200 range specified by the Harvard classification.This alignment underscores the precise and accurate delineation of spectral classifications within this specific type and catalog.

Confidence interval and testing hypothesis
To estimate the confidence interval for the population mean ( µ ), we utilize the sample mean ( x ) and the following equation (see 48 ) to determine the confidence intervals for each parameter in the EW systems: www.nature.com/scientificreports/ between period and T eff , as illustrated in Fig. 3 with an upward trend.Comparing our dataset with literature values (refer to Table 19), depicted in Figs. 2 and 3, indicates alignment with previously estimated values."

Discussion and conclusion
In this work we have presented a catalogue of ∼ 1800 EWs based on LAMOST, VSX and Gaia parameters.A details statistical analysis including: parameters distribution, confidence intervals and testing hypotheses to enable understanding the physical properties of such important eclipsing binary class.
In our catalog, we focused on several key parameters, including Period, Effective Temperature, Log(g), [Fe/H], and Radial Velocity, as well as the spectral type of the systems.Our study revealed that for EW systems, the mean period is 0.377 days and with 95% confidence, the majority falling within the range of 0.372 to 0.382 days.The mean effective temperature is approximately 5773 K, with most EW systems falling within the range of 5730 to 5820 K.The average metallicity is estimated to be − 0.185, and the majority of systems fall within the range of − 0.202 to − 0.168.The mean log of surface gravity for EW systems is approximately 4, with most samples ranging from 3.97 to 4.03.The average radial velocity for EW systems is − 4.085 km/s, within the range of − 6.47 to − 1.7 km/s.
Our study also confirms that the majority of EW systems are Late-type stars, primarily classified as F spectral type, followed by G and K.Among the sample, 88 systems are classified as A spectral type, with a mean surface temperature of 7400 K (i.e.stars with radiative envelopes).These findings could suggest that A-spectral type systems may not be classified as typical EW systems and they need a further investigations for better classifications.
To the best of our knowledge, this study represents the first instance of introducing confidence interval limits at a 95% confidence level for the atmospheric parameters of the EWs.Additionally, we conducted hypothesis testing based on these limits.However, prior research on general statistical properties of EWs has been undertaken by others, including studies by 20 and 55 .The authors in 20 focused on identifying peaks in the distribution of studied parameters and determined that the period, T eff , log(g), RV, and [Fe/H] exhibited peaks around 0.29 days, 5700 K, 4.16, − 20 km/s, and − 1.5, respectively.While our findings align with theirs for T eff , log(g), and [Fe/H], there are deviations in the observed periods and RV.Our study possesses the advantage of conducting a spectral type distribution analysis for EWs, leading to the conclusion that F-spectral types dominate among the various late-type systems.
On a different note 55 , collected data from approximately 700 previously analyzed systems to conduct a statistical investigation, focusing on parameters such as period, T eff , mass ratio, and the system's age.Their findings indicated that 50% of EWs have periods between 0.28 and 0.43 days, with a mean value of 0.35.Our results are  www.nature.com/scientificreports/comparable, as we observed that around 50% of our sample falls within periods ranging from 0.289 to 0.391, with a mean value of 0.34.They reported a mean T eff of approximately 5760 K, which closely matches our results (5770 K).
The correlation between the orbital period and the atmospheric parameters from the VSX and ZTF catalogs has been assessed.A strong agreement is observed, except for the period-Teff relation.Our findings indicate that ZTF periods align well with previously published relations, showing a correlation coefficient of 0.74.In contrast, a weak correlation is observed in the periods-Teff relation from the VSX catalog.This suggests a need for revising the VSX periods, as they may not be accurately recorded for the studied sample of EWs.In conclusion, our study enriches our understanding of Eclipsing W UMa systems by introducing confidence interval limits with hypothesis testing and focusing on spectral type distribution.This unique approach sets our work apart, providing a more comprehensive insight into this crucial class of eclipsing binaries.These findings not only advance our knowledge of EW systems but also open avenues for further investigations into their diverse characteristics, classifications, and evolutionary status.

Figure 1 .
Figure 1.Distribution of EW parameters.The X-axis represents the intervals of the parameter, while Y-axis is the number of Ew systems.

Figure 3 .
Figure 3. Period from ZTF catalog vs. Teff of the EW systems.

Table 1 .
Sample data for EW systems.

Table 2 .
Statistics of the studied parameters within our sample.

Table 3 .
Preiod frequency distribution.2% of the data.The remaining fraction of the dataset, totaling 210 EW systems, is distributed across the other nine metallicity intervals, constituting 11.8% of the sample.Figure1ddisplays the [Fe/H] distribution throughout the EW systems of the current sample.

Table 4 .
T eff frequency distribution.

Table 6 .
[Fe/H] frequency distribution.Radial velocity (RV)The "Radial Velocities Frequency Distribution" table reveals distinct patterns in the distribution of radial velocities (RV) for Eclipsing W Ursae Majoris (EW) systems.Over 51% of the EW systems are concentrated in the narrow RV range from − 54 to 3, while approximately 85% are clustered within adjacent categories from − 54 to 60.Moreover, about 92% of the EW systems are distributed in three categories spanning from − 111 to 60, with the remaining categories comprising 8% of the dataset (see, Table7).Notably, the majority of this 8% is found in the 60 to 117 category.Overall, around 97% of the EW systems fall within four categories ranging from -111 to 117 and beyond.

Table 8 .
Frequency distribution of A-spectral types.

Table 9 .
Proprieties of A spectral type.

Table 10 .
Frequency distribution of F spectral type.

Table 11 .
Proprieties of F spectral type.

Table 12 .
Frequency distribution of G-spectral types.

Table 13 .
Properties of G spectral type.

Table 14 .
Frequency distribution of K-spectral types.

Table 17 .
Correlation between the studied sample of EW's parameters.Periods are taken from VSX catalog.

Table 18 .
Correlation between the studied sample of EW's parameters.Periods are taken from ZTF catalog.

Table 19 .
Sample of studied EW systems with Period, T eff , and the corresponding Reference.